A method is described for deriving the sampling moments of means of random vectors obtained by sampling without replacement from a finite $k$-variate population of $n$ vector members. A table of results is presented listing the moments of order less than or equal to six as a function of the population moments. These moments were originally derived, in a less general form, in the course of developing the Simplex-Sum Designs discussed in [1]. Their possible wider applicability to sampling problems, however, motivated the extension of the work to the general formulas given here.

In Part A (Sections 1-5) the canonical forms of experiments concerning two simple hypotheses, and their partial ordering, are discussed. It is proved that every such experiment is a mixture (in a probability sense) of simple experiments whose sample spaces contain only two points. In Parts B (Sections 6-8) some general aspects of inference and decision problems are discussed in the usual theoretical framework, in which the overall mathematical model of an experiment is the frame of reference for all interpretations of outcomes. In Part C (Sections 9-16), attention is directed to that traditional function and basic problem of mathematical statistics, called here "informative inference," whose object is to recognize and report in appropriate objective terms those features of experimental outcomes which constitute statistical evidence relevant to hypotheses (or parameter values) of interest. The mathematical structure of statistical evidence and its qualitative and quantitative properties, are analyzed by application of (1) the mathematical results of Part A, which show that conditional experimental frames of reference (in the mixture sense) exist and are recognizable much more widely than has previously been realized; and (2) a single extra-mathematical proposition which many statisticians seem inclined to accept as appropriate for purposes of informative inference, a "principle of conditionality" which asserts that any outcome of any experiment which is a mixture of component experiments should be interpreted in the same way as if it were an outcome of just a corresponding component experiment (with the overall mixture structure otherwise ignored). This analysis establishes the likelihood function as the appropriate basis from which statistical inferences can be made directly without other reference to the structure of an experiment. For the numerical values of the likelihood function, this analysis provides direct interpretations in terms of probabilities of errors. These probabilities admit frequency interpretations of the usual kind, but they are not in general defined with reference to the specific experiment from which an outcome is obtained: they express intrinsic objective properties of the likelihood function itself, which this analysis shows to be appropriately relevant and directly useful for purposes of informative inference. The relations of this analysis of problems of informative inference to problems of testing statistical hypotheses, decision-making, conclusions, and Bayesian treatments of inference problems are discussed briefly. Generalizations of these mathematical results and their interpretations for problems involving more than two simple hypotheses will be given in a following paper.

Laplace ([13], p. 44, lines 5 and 6), in his statement concerning the "milieu de probabilite", seems to have referred to a probability distribution of the true value of a certain quantity ("le veritable instant du phenomene"), or, as we would say at present, to a probability distribution of a certain parameter. Thereby he differs from the attitude adopted in most of the work discussed in the present paper. Yet, one might hold that he possessed the idea of median-unbiased estimators. At any rate, when applying his notions to what Todhunter ([26] p. 469, art. 875) calls a case of no practical value, Laplace ([13], p. 48, lines 11 and 12 from the bottom) virtually rejected the use of arithmetic means of observations. Judging from innumerable texts, one finds that after him emphasis has long been mainly on mean-unbiasedness (see, however, Pitman (]20], bottom of p. 215), who mentions the existence of bias in the sense that the probability that a certain mean-unbiased estimator is less than the parameter in question is $>\frac{1}{2}$). Yet it is hard to find the requirement of mean-unbiasedness justified in print (cf., Brown ([3], lines 6-8 of Section 3): the average of independent mean-unbiased estimates is consistent; Lehmann ([14], lines 4-10 from bottom of p. 588): mean-unbiasedness flows from his general concept in the case of a quadratic loss function; Birnbaum ([2], p. 32): mean-unbiasedness is merely a technically useful property of the classical estimators in the linear estimation problem, which, at least in the case of normal errors, could equally well or preferably be justified on the basis of median-unbiasedness), much harder, in fact, than to find warnings against the hope that much is gained if an estimator be mean-unbiased (cf., Kendall ([12], Vol. 2, Section 17.9); the examples provided by Girshick, Mosteller and Savage ([9], middle of p. 20), Halmos ([10], the end of p. 43), Savage ([23], bottom of p. 244); lack of invariance under certain transformations being stressed by Halmos ([10], bottom of p. 42), Brown ([3], lines 13-16 of Section 3), Fisher ([7], p. 143, line 13 from bottom)). All the same, much interesting work has been devoted to mean-unbiased estimators, some of it investigating the conversion of biased estimators into unbiased ones (e.g., Quenouille [21], Olkin and Pratt [17]), or deriving unbiased estimators ab initio (e.g. Tate [25]). It is not the purpose of this paper to provide a bibliography that is at all near completeness, but it is interesting that the last two references illustrate a statement, made by Schmetterer ([24], middle of p. 215), to the effect that a close connection exists between integral equations and linear operators on the one hand, and the theory of mean-unbiased estimators on the other. This suggests that part of the motivation for the research in this field is of a mathematical, rather than a statistical nature. This view seems to be corroborated by Fraser's statement ([8], lines 12-14 from bottom of p. 49) to the effect that median-unbiasedness does not seem to lend itself to the mathematical analysis needed to find minimum risk estimates, and hence has found little application. The present paper seeks to extend the notion of unbiasedness (and the notion of bias) in a direction different from Lehmann [14] (who gave a definition within the framework of general decision theory), and from Brown [3] (who was primarily concerned with types of unbiasedness, among them median-unbiasedness, that are invariant "under simultaneous one-to-one transformations of the parameter and (its) estimate", or rather under simultaneous strictly monotone transformations of the parameter and its estimate), and from Peterson's [19] density-unbiasedness. It originated in work by the author [29], [30], on the estimation of the latent roots of certain matrices occurring in response surface theory. It had become clear that in this case it was of primary interest whether or not the frequency of obtaining too small (or too large, respectively) estimates would be unduly large. The present paper will make this notion more precise. Several types of bias (or of unbiasedness, respectively) will emerge, all of them clearly invariant in the sense of Brown. Median-unbiasedness will turn out to be a special case of this larger concept. Finally, certain seemingly unfamiliar properties of the sample median, of the product-moment correlation coefficient, and of Olkin and Pratt's function of the latter [17] will be proved and used to illustrate some of the concepts discussed.

A model which frequently arises from experimentation in psychology is one which contains both discrete and continuous variables. The concern in such a model may be with finding measures of association or with problems of inference on some of the parameters. In the simplest such model there is a discrete variable $x$ which takes the values 0 or 1, and a continuous variable $y$. Such a random variable $x$ is often used in psychology to denote the presence or absence of an attribute. Point-biserial correlation, which is the ordinary product-moment correlation between $x$ and $y$, has been used as a measure of association. This model, when $x$ has a binomial distribution, and the conditional distribution of $y$ for fixed $x$ is normal, was studied in some detail by Tate [13]. In the present paper, we consider a multivariate extension, in which $x = (x_0, x_1, \cdots, x_k)$ has a multinomial distribution, and the conditional distribution of $y = (y_1, \cdots, y_p)$ for fixed $x$ is multivariate normal.

This paper deals with confidence interval estimation and hypothesis testing for components of variance, in analysis-of-variance situations embraced by the Model II of Eisenhart [1], including also the so-called nested classifications. Several authors have treated the problem of setting confidence limits for variance components, and several approximate methods have been proposed. Four approximate methods are described in Anderson and Bancroft [2] and briefly in Crump [3], and references to original sources and extensive bibliographies are given in both [2] and [3]. In [4], Green gives more refined approximations which are, however, not presently in a form for practical use. Huitson [5], Welch [6] and Cochran [7] discuss related problems involving linear combinations of variances, and offer approximate methods for these problems. The many references on approximate tests and confidence limits in variance component problems emphasize the absence of exact methods. The present paper points out that, using a randomization device, exact confidence limits and tests for a variance component become available in a simple way. These exact confidence limits will usually but not always define a single confidence interval; in the exceptional case (having small probability in practice) the exact limits may define an interval with a gap in it. Numerical illustrations are given, together with comparisons with results using some of the available approximate methods. Also, asymptotic power comparisons between the exact test and two approximate tests are discussed. There are at least three notions of confidence that can be associated with the statement: "$a(x) \leqq \theta \leqq b(x)$ is a $100(1 - \alpha){\tt\%}$ confidence interval" for the parameter $\theta$ with possible nuisance parameters $\eta$, based on observations $x$. They are \begin{equation*}\begin{align*}\mathrm{(a)} \quad \operatorname{Pr}\{a(x) &\leqq \theta \leqq b(x)\} \equiv 1 - \alpha\quad \text{for all} \quad \theta, \eta \\ \mathrm{(b)} \quad \operatorname{Pr}\{a(x) &\leqq \theta \leqq b(x)\} \geqq 1 - \alpha\quad \text{for all} \quad \theta, \eta \quad \text{with equality for some} \quad \theta, \eta. \\ \mathrm{(c)} \quad \operatorname{Pr}\{a(x) &\leqq \theta \leqq b(x)\} \geqq 1 - \alpha \quad \text{for all} \quad \theta, \eta. \end{align*}\end{equation*} The phrase ``exact confidence" we shall interpret in the sense of (a) above. So far as the author is aware, for a variance component no confidence limits satisfying either of the notions (a) or (b) have previously been constructed. An interval satisfying (c) has been constructed in [8], using a two-stage sampling procedure. In the present approach, the mathematical difficulties ordinarily caused in variance component analysis by the presence of nuisance parameters are circumvented by a process of randomization. The resulting confidence limits depend not only on the mean squares of the analysis of variance table, but also on auxiliary observations on a random variable with known normal distribution. A consequence is that two statisticians confronted with the same analysis of variance table will in general construct different confidence limits. While practically this may afford some discomfiture, it remains true nevertheless that these exact confidence limits meet the ordinary claim (a) as to probability of containing the true variance component. In the examples tried, the limits are plausible and they are not difficult to compute. Moreover, the agreement between numerical results using the method proposed herein and the usual approximate methods may serve to increase one's faith in the approximate methods in small samples.

This paper considers the problem of finding confidence sets of the parallelepiped type based on extreme order statistics for multivariate medians when no parametric assumptions are made. A partial characterization of a multivariate distribution which will minimize the probability of the specified parallelepiped covering the multivariate median is given. This characterization enables one to obtain a sharp lower bound for the probability of coverage, provided the number of medians does not exceed seven and under the assumption that the structure is independent of the sample size.

Certain tests of independence based on the sample distribution function (d.f.) possess power properties superior to those of other tests of independence previously discussed in the literature. The characteristic functions of the limiting d.f.'s of a class of such test criteria are obtained, and the corresponding d.f. is tabled in the bivariate case, where the test is equivalent to one originally proposed by Hoeffding [4]. A discussion is included of the computational problems which arise in the inversion of characteristic functions of this type. Techniques for computing the statistics and for approximating the tail probabilities are considered.

I consider the calculation of the probability $P_n$ that the graph of a sample distribution function lie wholly to one side of a given arbitrary contour. A generating function approach is described in Section 2, and $P_n$ calculated exactly for some simple types of contour. Upper and lower bounds of the correct asymptotic form (relations (14), (15)) are obtained for $P_n$ in the case of a straight line contour.

Let $(R_{\nu 1}, \cdots, R_{{\nu N}_\nu})$ be a random vector which takes on the $N_\nu!$ permutations of $(1, \cdots, N_\nu)$ with equal probabilities. Let $\{b_{\nu i}, 1 \leqq i \leqq N_\nu, v \geqq 1\}$ and $\{a_{\nu i}, 1 \leqq i \leqq N_\nu, v \geqq 1\}$ be double sequences of real numbers. Put \begin{equation*}\tag{1.1}S_\nu = \sum^{N_\nu}_{i = 1} b_{\nu i}a_{\nu R_{\nu i}}.\end{equation*} We shall prove that the sufficient and necessary condition for asymptotic $(N_\nu \rightarrow \infty)$ normality of $S_\nu$ is of Lindeberg type. This result generalizes previous results by Wald-Wolfowitz [1], Noether [3], Hoeffding [4], Dwass [6], [7] and Motoo [8]. In respect to Motoo [8] we show, in fact, that his condition, applied to our case, is not only sufficient but also necessary. Cases encountered in rank-test theory are studied in more detail in Section 6 by means of the theory of martingales. The method of this paper consists in proving asymptotic equivalency in the mean of (1.1) to a sum of infinitesimal independent components.

This paper is concerned with the gap test for random sequences, first proposed by Kendall and Babington-Smith [7], and with various extensions to this test. One of these extensions is the test proposed by Meyer, Gephart and Rasmussen [8], another is, asymptotically, a partitioning of the $\chi^2$ statistic of Kendall and Babington-Smith [7], and others are likelihood ratio tests based on Markov chain models.

Let $S_m = X_1 + \cdots + X_m$, where the $X_j$ are independent random variables with common m.g.f. $\phi(t)$ which is assumed to exist in a real interval containing $t = 0$. Let the random variable $n$ be defined as the smallest integer $m$ for which either $S_m \geqq \alpha$ or $S_m \leqq - \beta(\alpha > 0, \beta > 0)$. Thus $n$ can be regarded as the time to absorption for the random walk $S_m$ with absorbing barriers at $\alpha$ and $-\beta$. Let $S = S_n$ and let $F_m(x) = P(-\beta < S_k < \alpha \quad \text{for}\quad k = 1, 2, \cdots m - 1 \quad \text{and} \quad S_m \leqq x)$. The main result of the paper is the identity \begin{equation*}\tag{0.1}E(e^{tS}z^n) = 1 + \lbrack z\phi (t) - 1\rbrack F(z, t),\end{equation*} where $F(z, t) = \sum^\infty_{m = 0} z^m \int^\alpha_{-\beta} e^{tx} dF_m(x).$ Wald's identity follows formally from (0.1) by setting $z = \lbrack\phi(t)\rbrack^{-1}$. Regions of validity of (0.1) and of Wald's identity are discussed, and it is shown that the latter holds for a larger range of values of $t$ than is usually supposed. In Section 5 there are three examples. In the first we consider the case where there is a single absorbing barrier and where the $X_j$ are discrete and bounded. This is a gambler's ruin problem, and we obtain an expression for the probability of ruin. In the second we use the classical random walk to illustrate the region of validity of (0.1). In the third we obtain the Laplace transform of the distribution of the time to absorption in a random walk in which steps of $+1$ and -1 occur at random in continuous time.

In this paper we shall investigate the so-called weak convergence of measures. Although the origin of the concept of the weak convergence of measures is a probabilistic one, the concept itself is purely measure-theoretical, and should be, therefore, treated by measure-theoretical methods. In Probability Theory the notion of the weak convergence of measures first appeared in Central Limit Problem. Its full importance, however, has been recognized only recently. It is now known as Donsker's Invariance Principle. In this paper we shall follow Prohorov's approach, as presented in [1]. The list of all necessary definitions and results is given in the Introduction. We shall give some conditions for the weak convergence of measures in separable and complete metric spaces, which are expressed in terms of convergence of measures generated in finite dimensional Euclidean spaces. The last convergence can be treated by standard mathematical tools, like the Theory of Fourier Transformations. It should be noted that our theorems concerning the convergence of measures in separable complete metric spaces remain valid if we omit the assumption of completeness. The proofs will remain essentially unchanged; only instead of dealing with compact sets, we should deal with totally bounded closed sets. The theorems given in Section 4 are of interest for the Theory of Stochastic Processes, since they give the conditions for the weak convergence of measures in the functional spaces $D\lbrack 0, 1\rbrack$ and $C\lbrack 0, 1\rbrack$, and to a large class of stochastic processes there correspond measures generated in space $C\lbrack 0, 1\rbrack$, and these measures are usually given in terms of $\mu^{t_1, \cdots, t_m}$, i.e. in terms of finite dimensional distribution functions of the process.

In a paper by Blackwell, Breiman and Thomasian [1, Theorem 3] the following theorem is proved: For any integer $n$ and for any $0 < \epsilon \leqq \frac{1}{2}$, such that $C - \epsilon \geqq 0$ there exists a code for a discrete memoryless channel with length $N > e^{n(C - \epsilon)}$ and with a bound for the probability of error, $\bar\lambda = 2 \exp_e - \lbrack n\epsilon^2/(16ab)\rbrack$, where $C$ is the capacity of the channel and a and b are the numbers of elements in the input and output alphabets respectively. In this note we shall replace the bound $2 \exp_e\lbrack -n\epsilon^2/(16ab)\rbrack$ by the expression $2 \exp_e\{-n\epsilon^2/\lbrack g(c)(\log c)^\{2 - \delta\rbrack\}$, where $c = \min (a, b), g(c)$ is a positive monotonically decreasing function of $c, g(c) < 16$ for all $c \geqq 3$ and approaches 2 asymptotically as $c \rightarrow \infty$, and $\delta > 0$ depends on $\epsilon$ and $c$ and tends to 0 as either $c \rightarrow \infty$ or $\epsilon \rightarrow 0$.

Consider a sequential decision problem in which independent observations are to be taken on a random variable $X$ whose distribution is of the form \begin{equation*}\tag{1}dG_\theta(x) = \Psi(\theta)e^{\theta x} d\mu(x),\end{equation*} where the parameter $\theta$ lies in a given interval $\Omega$ of the real line but is otherwise unknown, and the measure $\mu(x)$ is either absolutely continuous or discrete. The problem is to decide between the hypotheses \begin{equation*}\tag{2}\begin{split}H_1: \theta &\leqq \theta^\ast\\H_2: \theta &> \theta^\ast,\end{split}\end{equation*} where $\theta^\ast$ is a given point of $\Omega$. Under the assumptions stated in Section 2 the class A of generalized sequential probability ratio tests is shown to be essentially complete relative to the class D of decision functions with bounded risk. A decision function $\delta$ belongs to the class A if and only if after taking $n$ observations, (i) $\delta$ depends on the observations only through $n$ and $v_n = \sum^n_{i = 1} X_i$; (ii) $\delta$ specifies a closed interval $J_n : \lbrack a_{1n}, a_{2n}\rbrack$ for each $n$ and the following rule of action: (a) Stop experimentation as soon as $v_n \not\in J_n$. If $v_n < a_{1n}$, accept $H_1$. If $v_n > a_{2n}$, accept $H_2$. (b) If $a_{1n} < v_n < a_{2n}$, take another observation. (c) If $a_{1n} < a_{2n}$ and $v_n = a_{in}$, accept $H_i$ or take another observation or randomize between these two $(i = 1, 2)$. The problem considered here is the same as that treated by Sobel [1], and the foregoing statement of the problem and conclusion, as well as the assumptions to be given in the next section, follow his work very closely. The contribution of this paper is that the requirement of bounded loss functions made by Sobel is removed.